论文标题
弯曲位错的PEIERLS-NABARRO模型的有限稳定解的存在和唯一性
Existence and uniqueness of bounded stable solutions to the Peierls-Nabarro model for curved dislocations
论文作者
论文摘要
我们研究了矢量场peierls-nabarro模型的适当性，用于具有双井潜力的弯曲脱位，并且在远场限制了双态限制。使用Dirichlet到Neumann地图，3D PEIERLS-NABARRO模型还原为非本地标量Ginzburg-Landau方程。我们得出了非局部操作员的整体配方，当泊松比$ν\ in（ - \ frac12，\ frac13）$时，其内核是各向异性和积极的。然后，我们证明了该非本地标量金堡 - 兰道方程的任何有限的稳定解决方案具有1D曲线，这对应于具有各向异性非局部外周长的最小表面的PDE版本的平面度结果。基于此，我们最终获得了非本地标量表方程的稳态状态，以及原始的Peierls-Nabarro模型，可以将其表征为具有半laplacian的1D Ginzburg-Landau方程的单参数家族。
We study the well-posedness of the vector-field Peierls-Nabarro model for curved dislocations with a double well potential and a bi-states limit at far field. Using the Dirichlet to Neumann map, the 3D Peierls-Nabarro model is reduced to a nonlocal scalar Ginzburg-Landau equation. We derive an integral formulation of the nonlocal operator, whose kernel is anisotropic and positive when Poisson's ratio $ν\in(-\frac12, \frac13)$. We then prove that any bounded stable solutions to this nonlocal scalar Ginzburg-Landau equation has a 1D profile, which corresponds to the PDE version of flatness result for minimal surfaces with anisotropic nonlocal perimeter. Based on this, we finally obtain that steady states to the nonlocal scalar equation, as well as the original Peierls-Nabarro model, can be characterized as a one-parameter family of straight dislocation solutions to a rescaled 1D Ginzburg-Landau equation with the half Laplacian.
